// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#include "main.h"
#include "solverbase.h"
#include <Eigen/QR>
#include <Eigen/SVD>

template<typename MatrixType>
void
cod()
{
	STATIC_CHECK((internal::is_same<typename CompleteOrthogonalDecomposition<MatrixType>::StorageIndex, int>::value));

	Index rows = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
	Index cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
	Index cols2 = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
	Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);

	typedef typename MatrixType::Scalar Scalar;
	typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
	MatrixType matrix;
	createRandomPIMatrixOfRank(rank, rows, cols, matrix);
	CompleteOrthogonalDecomposition<MatrixType> cod(matrix);
	VERIFY(rank == cod.rank());
	VERIFY(cols - cod.rank() == cod.dimensionOfKernel());
	VERIFY(!cod.isInjective());
	VERIFY(!cod.isInvertible());
	VERIFY(!cod.isSurjective());

	MatrixQType q = cod.householderQ();
	VERIFY_IS_UNITARY(q);

	MatrixType z = cod.matrixZ();
	VERIFY_IS_UNITARY(z);

	MatrixType t;
	t.setZero(rows, cols);
	t.topLeftCorner(rank, rank) = cod.matrixT().topLeftCorner(rank, rank).template triangularView<Upper>();

	MatrixType c = q * t * z * cod.colsPermutation().inverse();
	VERIFY_IS_APPROX(matrix, c);

	check_solverbase<MatrixType, MatrixType>(matrix, cod, rows, cols, cols2);

	// Verify that we get the same minimum-norm solution as the SVD.
	MatrixType exact_solution = MatrixType::Random(cols, cols2);
	MatrixType rhs = matrix * exact_solution;
	MatrixType cod_solution = cod.solve(rhs);
	JacobiSVD<MatrixType> svd(matrix, ComputeThinU | ComputeThinV);
	MatrixType svd_solution = svd.solve(rhs);
	VERIFY_IS_APPROX(cod_solution, svd_solution);

	MatrixType pinv = cod.pseudoInverse();
	VERIFY_IS_APPROX(cod_solution, pinv * rhs);
}

template<typename MatrixType, int Cols2>
void
cod_fixedsize()
{
	enum
	{
		Rows = MatrixType::RowsAtCompileTime,
		Cols = MatrixType::ColsAtCompileTime
	};
	typedef typename MatrixType::Scalar Scalar;
	typedef CompleteOrthogonalDecomposition<Matrix<Scalar, Rows, Cols>> COD;
	int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols)) - 1);
	Matrix<Scalar, Rows, Cols> matrix;
	createRandomPIMatrixOfRank(rank, Rows, Cols, matrix);
	COD cod(matrix);
	VERIFY(rank == cod.rank());
	VERIFY(Cols - cod.rank() == cod.dimensionOfKernel());
	VERIFY(cod.isInjective() == (rank == Rows));
	VERIFY(cod.isSurjective() == (rank == Cols));
	VERIFY(cod.isInvertible() == (cod.isInjective() && cod.isSurjective()));

	check_solverbase<Matrix<Scalar, Cols, Cols2>, Matrix<Scalar, Rows, Cols2>>(matrix, cod, Rows, Cols, Cols2);

	// Verify that we get the same minimum-norm solution as the SVD.
	Matrix<Scalar, Cols, Cols2> exact_solution;
	exact_solution.setRandom(Cols, Cols2);
	Matrix<Scalar, Rows, Cols2> rhs = matrix * exact_solution;
	Matrix<Scalar, Cols, Cols2> cod_solution = cod.solve(rhs);
	JacobiSVD<MatrixType> svd(matrix, ComputeFullU | ComputeFullV);
	Matrix<Scalar, Cols, Cols2> svd_solution = svd.solve(rhs);
	VERIFY_IS_APPROX(cod_solution, svd_solution);

	typename Inverse<COD>::PlainObject pinv = cod.pseudoInverse();
	VERIFY_IS_APPROX(cod_solution, pinv * rhs);
}

template<typename MatrixType>
void
qr()
{
	using std::sqrt;

	STATIC_CHECK((internal::is_same<typename ColPivHouseholderQR<MatrixType>::StorageIndex, int>::value));

	Index rows = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE),
		  cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE),
		  cols2 = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
	Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);

	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;
	typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
	MatrixType m1;
	createRandomPIMatrixOfRank(rank, rows, cols, m1);
	ColPivHouseholderQR<MatrixType> qr(m1);
	VERIFY_IS_EQUAL(rank, qr.rank());
	VERIFY_IS_EQUAL(cols - qr.rank(), qr.dimensionOfKernel());
	VERIFY(!qr.isInjective());
	VERIFY(!qr.isInvertible());
	VERIFY(!qr.isSurjective());

	MatrixQType q = qr.householderQ();
	VERIFY_IS_UNITARY(q);

	MatrixType r = qr.matrixQR().template triangularView<Upper>();
	MatrixType c = q * r * qr.colsPermutation().inverse();
	VERIFY_IS_APPROX(m1, c);

	// Verify that the absolute value of the diagonal elements in R are
	// non-increasing until they reach the singularity threshold.
	RealScalar threshold = sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
	for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
		RealScalar x = numext::abs(r(i, i));
		RealScalar y = numext::abs(r(i + 1, i + 1));
		if (x < threshold && y < threshold)
			continue;
		if (!test_isApproxOrLessThan(y, x)) {
			for (Index j = 0; j < (std::min)(rows, cols); ++j) {
				std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
			}
			std::cout << "Failure at i=" << i << ", rank=" << rank << ", threshold=" << threshold << std::endl;
		}
		VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
	}

	check_solverbase<MatrixType, MatrixType>(m1, qr, rows, cols, cols2);

	{
		MatrixType m2, m3;
		Index size = rows;
		do {
			m1 = MatrixType::Random(size, size);
			qr.compute(m1);
		} while (!qr.isInvertible());
		MatrixType m1_inv = qr.inverse();
		m3 = m1 * MatrixType::Random(size, cols2);
		m2 = qr.solve(m3);
		VERIFY_IS_APPROX(m2, m1_inv * m3);
	}
}

template<typename MatrixType, int Cols2>
void
qr_fixedsize()
{
	using std::abs;
	using std::sqrt;
	enum
	{
		Rows = MatrixType::RowsAtCompileTime,
		Cols = MatrixType::ColsAtCompileTime
	};
	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;
	int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols)) - 1);
	Matrix<Scalar, Rows, Cols> m1;
	createRandomPIMatrixOfRank(rank, Rows, Cols, m1);
	ColPivHouseholderQR<Matrix<Scalar, Rows, Cols>> qr(m1);
	VERIFY_IS_EQUAL(rank, qr.rank());
	VERIFY_IS_EQUAL(Cols - qr.rank(), qr.dimensionOfKernel());
	VERIFY_IS_EQUAL(qr.isInjective(), (rank == Rows));
	VERIFY_IS_EQUAL(qr.isSurjective(), (rank == Cols));
	VERIFY_IS_EQUAL(qr.isInvertible(), (qr.isInjective() && qr.isSurjective()));

	Matrix<Scalar, Rows, Cols> r = qr.matrixQR().template triangularView<Upper>();
	Matrix<Scalar, Rows, Cols> c = qr.householderQ() * r * qr.colsPermutation().inverse();
	VERIFY_IS_APPROX(m1, c);

	check_solverbase<Matrix<Scalar, Cols, Cols2>, Matrix<Scalar, Rows, Cols2>>(m1, qr, Rows, Cols, Cols2);

	// Verify that the absolute value of the diagonal elements in R are
	// non-increasing until they reache the singularity threshold.
	RealScalar threshold = sqrt(RealScalar(Rows)) * (std::abs)(r(0, 0)) * NumTraits<Scalar>::epsilon();
	for (Index i = 0; i < (std::min)(int(Rows), int(Cols)) - 1; ++i) {
		RealScalar x = numext::abs(r(i, i));
		RealScalar y = numext::abs(r(i + 1, i + 1));
		if (x < threshold && y < threshold)
			continue;
		if (!test_isApproxOrLessThan(y, x)) {
			for (Index j = 0; j < (std::min)(int(Rows), int(Cols)); ++j) {
				std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
			}
			std::cout << "Failure at i=" << i << ", rank=" << rank << ", threshold=" << threshold << std::endl;
		}
		VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
	}
}

// This test is meant to verify that pivots are chosen such that
// even for a graded matrix, the diagonal of R falls of roughly
// monotonically until it reaches the threshold for singularity.
// We use the so-called Kahan matrix, which is a famous counter-example
// for rank-revealing QR. See
// http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
// page 3 for more detail.
template<typename MatrixType>
void
qr_kahan_matrix()
{
	using std::abs;
	using std::sqrt;
	typedef typename MatrixType::Scalar Scalar;
	typedef typename MatrixType::RealScalar RealScalar;

	Index rows = 300, cols = rows;

	MatrixType m1;
	m1.setZero(rows, cols);
	RealScalar s = std::pow(NumTraits<RealScalar>::epsilon(), 1.0 / rows);
	RealScalar c = std::sqrt(1 - s * s);
	RealScalar pow_s_i(1.0); // pow(s,i)
	for (Index i = 0; i < rows; ++i) {
		m1(i, i) = pow_s_i;
		m1.row(i).tail(rows - i - 1) = -pow_s_i * c * MatrixType::Ones(1, rows - i - 1);
		pow_s_i *= s;
	}
	m1 = (m1 + m1.transpose()).eval();
	ColPivHouseholderQR<MatrixType> qr(m1);
	MatrixType r = qr.matrixQR().template triangularView<Upper>();

	RealScalar threshold = std::sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
	for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
		RealScalar x = numext::abs(r(i, i));
		RealScalar y = numext::abs(r(i + 1, i + 1));
		if (x < threshold && y < threshold)
			continue;
		if (!test_isApproxOrLessThan(y, x)) {
			for (Index j = 0; j < (std::min)(rows, cols); ++j) {
				std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
			}
			std::cout << "Failure at i=" << i << ", rank=" << qr.rank() << ", threshold=" << threshold << std::endl;
		}
		VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
	}
}

template<typename MatrixType>
void
qr_invertible()
{
	using std::abs;
	using std::log;
	typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
	typedef typename MatrixType::Scalar Scalar;

	int size = internal::random<int>(10, 50);

	MatrixType m1(size, size), m2(size, size), m3(size, size);
	m1 = MatrixType::Random(size, size);

	if (internal::is_same<RealScalar, float>::value) {
		// let's build a matrix more stable to inverse
		MatrixType a = MatrixType::Random(size, size * 2);
		m1 += a * a.adjoint();
	}

	ColPivHouseholderQR<MatrixType> qr(m1);

	check_solverbase<MatrixType, MatrixType>(m1, qr, size, size, size);

	// now construct a matrix with prescribed determinant
	m1.setZero();
	for (int i = 0; i < size; i++)
		m1(i, i) = internal::random<Scalar>();
	RealScalar absdet = abs(m1.diagonal().prod());
	m3 = qr.householderQ(); // get a unitary
	m1 = m3 * m1 * m3;
	qr.compute(m1);
	VERIFY_IS_APPROX(absdet, qr.absDeterminant());
	VERIFY_IS_APPROX(log(absdet), qr.logAbsDeterminant());
}

template<typename MatrixType>
void
qr_verify_assert()
{
	MatrixType tmp;

	ColPivHouseholderQR<MatrixType> qr;
	VERIFY_RAISES_ASSERT(qr.matrixQR())
	VERIFY_RAISES_ASSERT(qr.solve(tmp))
	VERIFY_RAISES_ASSERT(qr.transpose().solve(tmp))
	VERIFY_RAISES_ASSERT(qr.adjoint().solve(tmp))
	VERIFY_RAISES_ASSERT(qr.householderQ())
	VERIFY_RAISES_ASSERT(qr.dimensionOfKernel())
	VERIFY_RAISES_ASSERT(qr.isInjective())
	VERIFY_RAISES_ASSERT(qr.isSurjective())
	VERIFY_RAISES_ASSERT(qr.isInvertible())
	VERIFY_RAISES_ASSERT(qr.inverse())
	VERIFY_RAISES_ASSERT(qr.absDeterminant())
	VERIFY_RAISES_ASSERT(qr.logAbsDeterminant())
}

template<typename MatrixType>
void
cod_verify_assert()
{
	MatrixType tmp;

	CompleteOrthogonalDecomposition<MatrixType> cod;
	VERIFY_RAISES_ASSERT(cod.matrixQTZ())
	VERIFY_RAISES_ASSERT(cod.solve(tmp))
	VERIFY_RAISES_ASSERT(cod.transpose().solve(tmp))
	VERIFY_RAISES_ASSERT(cod.adjoint().solve(tmp))
	VERIFY_RAISES_ASSERT(cod.householderQ())
	VERIFY_RAISES_ASSERT(cod.dimensionOfKernel())
	VERIFY_RAISES_ASSERT(cod.isInjective())
	VERIFY_RAISES_ASSERT(cod.isSurjective())
	VERIFY_RAISES_ASSERT(cod.isInvertible())
	VERIFY_RAISES_ASSERT(cod.pseudoInverse())
	VERIFY_RAISES_ASSERT(cod.absDeterminant())
	VERIFY_RAISES_ASSERT(cod.logAbsDeterminant())
}

EIGEN_DECLARE_TEST(qr_colpivoting)
{
	for (int i = 0; i < g_repeat; i++) {
		CALL_SUBTEST_1(qr<MatrixXf>());
		CALL_SUBTEST_2(qr<MatrixXd>());
		CALL_SUBTEST_3(qr<MatrixXcd>());
		CALL_SUBTEST_4((qr_fixedsize<Matrix<float, 3, 5>, 4>()));
		CALL_SUBTEST_5((qr_fixedsize<Matrix<double, 6, 2>, 3>()));
		CALL_SUBTEST_5((qr_fixedsize<Matrix<double, 1, 1>, 1>()));
	}

	for (int i = 0; i < g_repeat; i++) {
		CALL_SUBTEST_1(cod<MatrixXf>());
		CALL_SUBTEST_2(cod<MatrixXd>());
		CALL_SUBTEST_3(cod<MatrixXcd>());
		CALL_SUBTEST_4((cod_fixedsize<Matrix<float, 3, 5>, 4>()));
		CALL_SUBTEST_5((cod_fixedsize<Matrix<double, 6, 2>, 3>()));
		CALL_SUBTEST_5((cod_fixedsize<Matrix<double, 1, 1>, 1>()));
	}

	for (int i = 0; i < g_repeat; i++) {
		CALL_SUBTEST_1(qr_invertible<MatrixXf>());
		CALL_SUBTEST_2(qr_invertible<MatrixXd>());
		CALL_SUBTEST_6(qr_invertible<MatrixXcf>());
		CALL_SUBTEST_3(qr_invertible<MatrixXcd>());
	}

	CALL_SUBTEST_7(qr_verify_assert<Matrix3f>());
	CALL_SUBTEST_8(qr_verify_assert<Matrix3d>());
	CALL_SUBTEST_1(qr_verify_assert<MatrixXf>());
	CALL_SUBTEST_2(qr_verify_assert<MatrixXd>());
	CALL_SUBTEST_6(qr_verify_assert<MatrixXcf>());
	CALL_SUBTEST_3(qr_verify_assert<MatrixXcd>());

	CALL_SUBTEST_7(cod_verify_assert<Matrix3f>());
	CALL_SUBTEST_8(cod_verify_assert<Matrix3d>());
	CALL_SUBTEST_1(cod_verify_assert<MatrixXf>());
	CALL_SUBTEST_2(cod_verify_assert<MatrixXd>());
	CALL_SUBTEST_6(cod_verify_assert<MatrixXcf>());
	CALL_SUBTEST_3(cod_verify_assert<MatrixXcd>());

	// Test problem size constructors
	CALL_SUBTEST_9(ColPivHouseholderQR<MatrixXf>(10, 20));

	CALL_SUBTEST_1(qr_kahan_matrix<MatrixXf>());
	CALL_SUBTEST_2(qr_kahan_matrix<MatrixXd>());
}
